Program for the Year 2019 to 20 | Spectral Geometry in the Clouds

Program of the seminar for the Academic Year 2019 to 2020

April 20	Michael Levitin (University of Reading)
	Inverse Steklov spectral problem for curvilinear polygons
	View Abstract I will present results of a recent preprint, joint with Stanislav Krymski, Leonid Parnovski, Iosif Polterovich, and David Sher. For a generic curvilinear polygon with angles less than pi, we prove that the asymptotics of Steklov eigenvalues obtained previously in arXiv:1908.06455, determines, in a constructive manner, the number of vertices, and the properly ordered sequence of side lengths, as well as the angles of the polygon, up to a certain equivalence relation. I will also present counterexamples to this statement if the generic assumptions fail.
April 27	Emilio Lauret (Universidad Nacional de Córdoba)
	On the first Laplace eigenvalue of a homogeneous sphere
	View Abstract We will show an explicit expression for the lowest positive eigenvalue of the Laplace-Beltrami operator associated to any homogeneous sphere (a sphere endowed with a Riemannian metric such that the action of its isometry group on it is transitive). The expression is in terms of the parameters defining the homogeneous metric. As a consequence, we will show that the Laplace spectrum distinguishes any metric among homogeneous spheres, and furthermore, we will obtain some applications to the Yamabe problem. Part of these results is in collaboration with Renato Bettiol and Paolo Piccione.
May 4	Dorin Bucur (Université de Savoie)
	Stable and unstable spectral inequalities
	View Abstract In the last years, the stability question of several classical spectral inequalities has been raised in the vein of the result by Fusco, Maggi and Pratelli (2008), which gives a sharp quantitative form of the isoperimetric inequality. After an introduction to this topic, I will focus on some recent results obtained with M. Nahon and A. Giacomini on spectral problems involving boundary energies. Precisely, following a question raised by Girouard and Polterovich, I will show that the Weinstock inequality is genuinely unstable, namely that the supremum of the (perimeter normalized) first non-zero eigenvalue of the Steklov problem can be achieved in the geometric neighbourhood of any smooth simply connected domain of the plane. Time remaining, I will introduce a new method to prove quantitative forms of spectral inequalities of Robin type which relies on the analysis of a new class of geometric/energy functionals in the context of free discontinuity/free boundary problems. This talk is a based on joint works with M. Nahon and A. Giacomini.
May 11	Henrik Matthiesen (University of Chicago)
	Asymptotic behaviour of eigenvalues in glueing constructions and new minimal surfaces via shape optimization
	View Abstract Since maximizing metrics for the Laplace eigenvalues, normalized by the area, and the Steklov eigenvalues, normalized by the length of the boundary, arise as induced metrics from minimal surfaces, we can try to solve these optimization problems to find minimal surface. In the case of the Laplace operator on closed surfaces these are minimal surfaces in round spheres. For Steklov eigenvalues on surfaces with boundary these are free boundary minimal surfaces in Euclidean balls. Very recently, these maximization problems have been fully settled in both cases for the first eigenvalue. In particular, this gives rise to new examples of minimal surfaces of the aforementioned types. I will first give a very brief sketch of the program to obtain these maximizing metrics. Then, I will try to give some idea of our glueing constructions completing this program for the first eigenvalue. This is based on joint work with Anna Siffert and joint work with Romain Petrides.
May 18	Yaiza Canzani (UNC Chapel Hill)
	$L^p$-norms via geodesic beams
	View Abstract In this talk, we discuss a geodesic beams approach for understanding eigenfunction concentration, and give applications to the study of the $L^p$ norms of eigenfunctions for $p$ larger than the critical exponent. In particular, we present quantitative improvements on the standard estimates which rely only on dynamical assumptions. This is joint work with J. Galkowski.
May 25	Iosif Polterovich (Université de Montréal)
	Lambdas, bubbles, and spheres
	View Abstract Geometric optimization of Laplace eigenvalues on surfaces under the area constraint is closely linked to the study of minimal surfaces in spheres and harmonic maps. I will discuss some recent developments in the subject, including a solution of the maximization problem for higher eigenvalues on the 2-sphere. The talk is based on joint works with M. Karpukhin, N. Nadirashvili and A. Penskoi.
June 1	Xuwen Zhu (MIT)
	Spectral properties of reducible spherical conical metrics
	View Abstract This talk will focus on the recent work, joint with Rafe Mazzeo and Bin Xu, on the spectral properties of constant curvature metrics with conical singularities on surfaces. We will focus on a special class called spherical conical metrics with reducible monodromy. We obtain a spectral characterization of the monodromy property, which establishes a new connection between the geometric microlocal analysis which is about the study of singular operators, and the classical complex analysis approach that uses the developing maps. Those spectral properties help us to understand the moduli spaces of spherical conical metrics, which have seen a lot of recent development and yet there are still a lot of open problems.
June 8	Peter Humphries (University College London)
	The random wave conjecture and arithmetic quantum chaos
	View Abstract Berry's random wave conjecture is a heuristic that the eigenfunctions of a classically ergodic system ought to display Gaussian random behaviour, as though they were random waves, in the large eigenvalue limit. We discuss two aspects of this problems for eigenfunctions of the Laplacian on a particular number-theoretic negatively curved surface: Planck scale mass equidistribution, and an asymptotic for the fourth moment.
June 15	Cristina Trombetti (Università degli Studi di Napoli Federico II)
	Nodal Sets of Eigenfunctions: Progress via Optimal Transport
	View Abstract Comparison results (of isoperimetric type) for solutions to elliptic equations with Dirichlet and Neumann boundary conditions have been established since decades. So far not many results have been obtained in the case of Robin boundary conditions. In this talk we investigate some open questions related to a Faber Krahn inequality and Talenti-type estimates.
June 22	Stefan Steinerberger (Yale)
	Nodal Sets of Eigenfunctions: Progress via Optimal Transport
	View Abstract An old and important question about Laplacian eigenfunctions is to understand the size of the set where they vanish (their nodal set): the idea is that eigenfunctions corresponding to larger frequencies should oscillate more and thus vanish on a larger set. The same principle should then also be true for linear combinations of high-frequency eigenfunctions (in one dimension, this follows from Sturm-Liouville theory). Recent progress on this question is based on the notion of optimal transport and a very simple idea which we formalize: if it's easy to buy milk, then there must be many supermarkets (and, conversely, if there are only few supermarkets some people have to travel a large distance to buy milk). This turns into a geometric inequality that is interesting in its own right. The methods are self-contained and elementary but many open problems remain.
June 29	Daniel Stern (University of Toronto)
	Min-max harmonic maps and extremal metrics for Laplacian eigenvalues
	View Abstract I'll describe recent work with Mikhail Karpukhin, in which we relate the problem of maximizing Laplacian eigenvalues over unit-area metrics on a given Riemann surface to natural constructions of harmonic maps to high-dimensional spheres, via min-max methods for the Dirichlet energy. I'll explain how our methods give a new construction of conformal metrics maximizing the first and second Laplacian eigenvalues, and yield new estimates for other shape optimization problems in spectral geometry — for instance providing asymptotically sharp upper bounds for the first two Steklov eigenvalues on surfaces with fixed genus and any number of boundary components.
July 6	Rupert Frank (Caltech)
	Sharp Weyl Laws in 3d with rough potentials
	View Abstract We consider the Laplace-Beltrami operator on a three-dimensional Riemannian manifold perturbed by a rough potential from the Kato class and study whether various forms of Weyl's law remain valid under this perturbation. In particular, we show that the pointwise Weyl law with sharp remainder estimate is valid under an assumption that is slightly stronger than Kato class and that, on the other hand, this sharp remainder estimate can be violated for Kato class potentials. For the proof we extend the method of Avakumović to the case of Schrödinger operators with singular potentials. The talk is based on joint work with J. Sabin.
July 13	Panagiotis Polymerakis (Max Planck Institute for Mathematics)
	Bottom of spectra and coverings
	View Abstract In this talk, we will discuss some results on the behaviour of the spectrum of Schrödinger operators under Riemannian coverings. In particular, we will focus on the relation between the amenability of a covering and the preservation of the bottom of the spectrum.
July 20	Asma Hassannezhad (University of Bristol)
	Eigenvalue bounds for the mixed Steklov problem
	View Abstract There have been many developments on Steklov eigenvalue bounds over the last few years. We will discuss how some of these bounds can be extended and improved for the mixed Steklov problem, in particular, for the Steklov-Neumann problem. These bounds include Hersch-Payne-Schiffer inequality and Weinstock inequality. The Steklov-Neuman eigenvalues problem naturally appears when we study the Steklov problem on Riemannian orbifolds, and we also consider our results in this setting. The talk is mainly based on joint work with T. Arias-Marco, E. Dryden, C. Gordon, A. Ray and E. Stanhope.
July 27	Julie Rowlett (Chalmers University of Technology)
	Same-same but different: Polyakov formulas via microlocal analysis and mathematical physics
	View Abstract The zeta-regularized determinant of the Laplacian connects analysis, geometry, number theory, and physics. It is simultaneously beautiful and elusive, because in general it is impossible to compute in closed form, due to its global nature. To glean information, many have considered the variation of the determinant, because in contrast, this is a local quantity. Here, we shall investigate the variation of the determinant in singular geometric contexts. We will discuss so-called variational Polyakov formulas for surfaces and domains with singularities and boundaries. For the special cases of finite circular sectors and finite cones, we will show two different methods for obtaining the variation of the determinant. At first glance, the formulas obtained via microlocal analysis versus mathematical physics look quite different. It is an utter joy that although they appear different, we can prove that the expressions are indeed the same. This talk is based on joint work with Clara Aldana and Klaus Kirsten.
August 3-4	Mini Conference: Young researchers in spectral geometry
August 3 8AM PDT 11AM EDT 4PM BST 5 PM CEST	Matteo Capoferri (Cardiff University)
	Hyperbolic propagators and invariant subspaces of elliptic systems
	View Abstract In my talk I will report on recent results about the spectral theory of elliptic systems on closed manifolds. In the first part of the talk, I will present an explicit (i.e. up to solving ODEs), global (in space and in time) and invariant (under changes of local coordinates and gauge transformations that may be present) construction of wave propagators for first order systems, with a particular focus on the Dirac operator on a closed 3-manifold. The core of our approach is the theory of global Fourier integral operators with complex-valued phase functions developed by Laptev, Safarov and Vassiliev in the nineties. In the second part of the talk, we will consider an elliptic pseudodifferential operator $A$ of arbitrary positive order acting on $m$-columns of half-densities and discuss how one can partition its eigenvalues in precisely $m$ (infinite) families, one for each eigenvalue of the principal symbol of $A$. Our results, which rely on the construction of an orthonormal pseudodifferential basis, allow us to refine two-terms asymptotic formulae for the eigenvalue counting function of $A$ available in the literature. Furthermore, such a decomposition of the spectrum of $A$ provides some additional insight into the above propagator construction. Joint work with Dima Vassiliev (UCL).
August 3 9AM PDT 12AM EDT 5PM BST 6 PM CEST	Ingrid Membrillo Solis (University of Southampton)
	Hearing 2 and 3 orbifolds with heat invariants
	View Abstract A Riemannian orbifold is a generalisation of a manifold in which the local structure is that of a quotient of a Euclidean space by the action of a finite group of isometries. Such a local structure allows using the tools of spectral geometry, including the Laplace operator and its spectrum. In this context, a natural question is whether spectral data detects the presence of orbifold singularities. In 2008 Dryden, Gordon, Greenwald and Webb gave a partial affirmative answer using orbifold heat invariants for the Laplace operator acting on functions, however, there were some remaining cases. In this talk I will discuss the computation of heat invariants for the Hodge Laplacian acting on 1-forms on Riemmanian orbifolds. As an application, I will show how a combination of the heat invariants for functions along with those for 1-forms allows us to distinguish a singular $n$-dimensional orbifold from an $n$-dimensional manifold for $n=2,3$. This is a joint work with K. Gittins, C. Gordon, M. Khalile, M. Sandoval and E. Stanhope.
August 3 10AM PDT 1PM EDT 6PM BST 7 PM CEST	Hadrian Quan (University of Illinois)
	The Heat Kernel of a Contact Manifold in the Sub-Riemannian Limit
	View Abstract In this talk I will describe joint work with Pierre Albin in which we study the limits of heat kernels on contact manifolds corresponding to a Riemannian metric degenerating to a submetric of the contact structure. This question is approached using the tools of geometric microlocal analysis. Time permitting, I will discuss connections with the Rumin complex, and the associated limit of Analytic Torsion.
August 4 8AM PDT 11AM EDT 4PM BST 5 PM CEST	Corentin Léna (Stockholm University)
	Variations on Pleijel’s nodal theorem
	View Abstract Courant’s nodal theorem tells us that an eigenfunction of the Laplacian associated with eigenvalue number $k$ has at most $k$ nodal domains. Å. Pleijel showed in 1956 that for a given planar domain, with a Dirichlet boundary condition, equality can be reached only for a finite number of eigenvalues. Pleijel’s proof actually gives an asymptotic upper bound of the number of nodal domains. It has been extended afterwards to other geometric settings, boundary conditions and operators. The topic has received a lot of attention in the past decade and substantial progress has been made. Several generalizations and refined versions have been obtained, and a large number of special cases analyzed. I will describe these recent developments and the many open questions in this area. This will include joint work with K. Gittins.
August 4 9AM PDT 12AM EDT 5PM BST 6 PM CEST	Samuel Lin (Dartmouth College)
	Three-dimensional Geometric Structures and the Laplace Spectrum
	View Abstract The earliest examples of non-isometric Laplace-isospectral manifolds have the same local geometries. In fact, the first example of 16-tori given by Milnor and other isospectral pairs arising from the classical group theoretic method of Sunada have the same local geometries. However, examples from Gordon, Schueth, Sutton, and An-Yu-Yu demonstrate that in dimension four and higher, the local geometry is not a spectral invariant, even among locally homogeneous spaces. Thus it is natural to ask whether the local geometry is a spectral invariant in dimension two and three. I will present our result in this direction, which provides strong evidence that the local geometry of a three-dimensional locally homogeneous space is a spectral invariant. This talk is based on a joint work with Ben Schmidt and Craig Sutton.
August 4 10AM PDT 1PM EDT 6PM BST 7 PM CEST	Philippe Charron (Université de Montréal)
	Critical points of Laplace eigenfunctions
	View Abstract In this talk, we will review some recent (and not so recent) results concerning the number of critical points of Laplace eigenfunctions on manifolds. We will outline the construction of smooth metrics on the torus and on the sphere such that certain eigenfunctions have either very few or infinitely many critical points. We will also discuss some stability results. This talk includes joint work with Pierre Bérard and Bernard Helffer.
August 24	Special seminar on the occasion of Dima Jakobson's 50th birthday
8 PDT 11 EDT 16 BST 17 CEST	Yaiza Canzani (University of North Carolina at Chapel Hill)
	Weyl remainders: an application of geodesic beams
	View Abstract In this talk we discuss quantitative improvements for Weyl remainders under dynamical assumptions on the geodesic flow. We consider a variety of Weyl type remainders including asymptotics for the eigenvalue counting function as well as for the on and off-diagonal spectral projector. These improvements are obtained by combining the geodesic beam approach to understanding eigenfunction concentration together with an appropriate decomposition of the spectral projector into quasimodes for the Laplacian. One consequence of these estimates is a quantitatively improved Weyl remainder for the eigenvalue counting function that holds for all product manifolds. This is joint work with J. Galkowski.
9 PDT 12 EDT 17 BST 18 CEST	Mikhail Karpukhin (California Institute of Technology)
	Index of minimal surfaces in spheres and eigenvalues of the Laplacian
	View Abstract The Laplacian is a canonical second order elliptic operator defined on any Riemannian manifold. The study of upper bounds for its eigenvalues is a classical problem of spectral geometry going back to J. Hersch, P. Li and S.-T. Yau. It turns out that the optimal isoperimetric inequalities for Laplacian eigenvalues are closely related to minimal surfaces in spheres. At the same time, the index of a minimal surface is defined as a number of negative eigenvalues of a different second order elliptic operator. It measures the instability of the surface as a critical point of the area functional. In the present talk we will discuss the interplay between index and Laplacian eigenvalues, and present some recent applications, including a new bound on the index of minimal spheres as well as the optimal isoperimetric inequality for Laplacian eigenvalues on the projective plane.
11 PDT 14 EDT 19 BST 20 CEST	Tadashi Tokieda (Stanford)
	Applying physics to mathematics
	View Abstract Humans tend to be better at physics than at mathematics. When an apple falls from a tree, there are more people who can catch it—they physically know how the apple moves—than people who can compute its trajectory from a differential equation. Applying physical ideas to discover and prove mathematical results is therefore natural, even if it has seldom been tried in the history of science. (The exceptions include Archimedes, some old Russian sources, a recent book of Levi's, as well as my articles and lectures.) A variety of examples will be presented.